Probability and Conditional Probability: What Changes When You Learn More?

Probability measures how likely an event is, from 0% to 100%. Conditional probability measures how likely it is given a specified condition. The distinction matters when new information arrives—and when a market price seems lower than your forecast.

Write P(A) for the probability of event A and P(A | B) for the probability of A given B. The vertical bar means “given,” not “and.”

Define the outcome before estimating it

Consider a hypothetical Yes/No market:

Will Station X record at least 1.0 mm of rain between noon and midnight on October 12, 2026, according to its official report for that interval?

Assume its rules specify the time zone and what happens if the report is missing. For this example, assume the report will be available. Let A mean that the contract resolves Yes—not merely that someone nearby sees rain.

That precision matters. Polymarket’s resolution documentation, for example, explains that market rules identify the resolution source, end date and treatment of edge cases. Your estimate needs to match those rules.

Now let B mean that cloud cover at Station X is at least 80% at noon. B is information that might help forecast A; it is not an extra requirement for a Yes payout.

One example: change the denominator

Suppose your model contains 100 equally likely scenarios:

Noon condition	Rain threshold met: A	Threshold not met	Total
Cloud cover at least 80%: B	42	18	60
Cloud cover below 80%: not B	8	32	40
Total	50	50	100

These are invented model assumptions, not weather observations.

Before learning the noon cloud condition, 50 of the 100 scenarios meet the rain threshold:

P(A) = 50 / 100 = 50%.

If B is confirmed, restrict attention to the 60 scenarios in the first row. Of those, 42 meet the threshold:

P(A | B) = 42 / 60 = 70%.

The general formula is:

P(A | B) = P(A and B) / P(B), provided P(B) > 0.

Conditioning means excluding outcomes outside B and recalculating within what remains, as explained in this probability textbook’s conditional-probability section.

The update from 50% to 70% does not make rain certain. Under the same assumptions, the chance of failing to meet the threshold given B is 18 / 60 = 30%.

“Given,” “and” and the reverse question differ

The same table answers three different questions:

• Rain given high cloud cover: P(A | B) = 42 / 60 = 70%.
• Rain and high cloud cover: P(A and B) = 42 / 100 = 42%.
• High cloud cover given rain: P(B | A) = 42 / 50 = 84%.

You cannot assume that reversing “given” preserves the probability: the denominator changes.

Nor should you multiply the two standalone probabilities automatically. Here, 50% × 60% = 30%, not the correct joint probability of 42%. Multiplying P(A) by P(B) works for independent events—events for which learning B does not change A’s probability. See the textbook’s definition of independence.

For this example, use P(A and B) = P(B) × P(A | B) = 60% × 70% = 42% instead.

Also distinguish a confirmed condition from a forecast. “High cloud cover is expected” does not establish B. While the cloud condition remains unknown, your rain estimate must account for both cloud-cover possibilities.

A 70% estimate is not automatically a purchase

Continue the example after B is confirmed. Suppose the Yes order book has:

• Best bid: 66¢
• Best ask: 70¢
• Midpoint: 68¢
• Spread: 4¢

The midpoint is a reference, not the price available for an immediate purchase. Polymarket’s order-book documentation distinguishes the lowest ask for buying, highest bid for selling and their midpoint.

Assume one contract pays $1 for Yes and $0 for No, you hold through resolution, sufficient quantity is available at 70¢, and total fees are a hypothetical 2¢ per contract. The $1/$0 structure matches the winning and losing token payouts described by Polymarket.

Then:

Expected payout = 70% × $1 = $0.70.

All-in cost = $0.70 + $0.02 = $0.72.

Expected net result = $0.70 − $0.72 = −$0.02 per contract.

Under these assumptions, your break-even probability is 72%; the 68¢ midpoint is not your all-in purchase cost. The spread is already reflected in using the ask; do not add it again as another fee.

Before acting, check what your probability describes, whether its condition is actually known, and whether the executable cost fits that estimate. The broader pre-trade checklist covers the remaining market checks. This calculation is education, not a recommendation to trade.